On the D Calculus for Linear Differential Equations with Constant Coefficients

Extract

1. Introduction. While the operational calculus for the solution o initial value problems for linear differential equations with constam coefficients is now commonly introduced in a thoroughly satisfactory manner—either by means of the Laplace transform (e.g. ref. 1) of by the method of J- Mikusinski (ref. 2)—the same cannot be said of the simpler method for the general integration of such equations which is known as the D calculus. In fact, I cannot recall any text which gives a consistent definition of the meaning of expressions of the type [F(D)/G(D)]y, where F and G are polynomials. The reason for this state of affairs seems to be that the scope of the D calculus is rather limited so that the validity of the result can be verified in each case (e.g. ref. 3). In particular, this applies to the decomposition of an operator F-¹(D) into partial fractions, which is the central step: in the solution of an equation with nonvanishing right hand side of general form. Nevertheless a rational approach to the entire problem is perhaps not out of place. This is attempted in the present paper.